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In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number.

For example, the logarithm of 1000 to the base 10 is 3, because 10 raised to the power of 3 is 1000. As another example, the logarithm of 32 to the base 2 is 5, because 2 raised to the power 5 is 32.

The logarithm of $x$ to the base $b$ is written $\log _{b}\,(x)$ or, if the base is implicit, as $\log(x)$. So, for a number $x$, a base $b$ and an exponent $y$:

${\text{if }}b^{y}=x,{\text{ then }}y=\log _{b}\,(x)$

The $\log _{b}\,(x)$ is a unique real number when $x$ and $b$ are restricted to positive real numbers and is negative for $0<b<1$, zero for $b=1$, and positive for $b>1$.

Features of the logarithm

An important feature of logarithms is that they reduce multiplication to addition. That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers as in this identity:

$\log(xy)=\log x+\log y$

Logarithms also reduce division to subtraction as in this identity:

$\log(x/y)=\log(x)-\log(y)$

And they reduce exponation to multiplication as in this identity:

$\log(x^{y})=y\log(x)$

And taking roots are reduced to division:

$\log({\sqrt[{y}]{x}})={\frac {\log(x)}{y}}$

The inverse of the logarithm is call the antilogarithm and it is expressed in this identity:

${\text{antilog}}_{b}(n)=b^{n}$

Although the above practical advantages are not important for
 numerical work today, they are used in graphical analysis (see Bode plot).

Bases and notations

The base $b$ must not be 0 nor 1. The most widely used bases for logarithms are 10, the mathematical constant e ≈ 2.71828 (also known as "Euler's number") and 2. When "log" is written without a base (b missing from log_b), the intent can sometimes be determined from the context in which it used:

• natural logarithm (log_e, ln, log, or Ln) in mathematical analysis, statistics, economics and some engineering fields. The reasons to consider e the natural base for logarithms, though perhaps not obvious, are numerous and compelling.
• common logarithm (log₁₀ or simply log; sometimes lg) in various engineering fields, especially for power levels and power ratios, such as acoustical sound pressure, and in logarithm tables to be used to simplify hand calculations
• binary logarithm (log₂; sometimes lg, lb, or ld), in computer science and information theory
• indefinite logarithm (Log or [log ] or simply log) when the base is irrelevant, e.g. in complexity theory when describing the asymptotic behavior of algorithms in big O notation.

To avoid confusion, it is always best to specify the base if there is any chance of misinterpretation.

The notation "ln(x)" invariably means log_e(x), i.e., the natural logarithm of x, but the implied base for "log(x)" varies by discipline:

• Mathematicians understand "log(x)" to mean log_e(x). Calculus textbooks will occasionally write "lg(x)" to represent "log₁₀(x)".

• Many engineers, biologists, astronomers, and some others write only "ln(x)" or "log_e(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log₁₀(x) or, in computer science, log₂(x).

• On most calculators, the LOG button is log₁₀(x) and LN is log_e(x).

• In most commonly used computer programming languages, including C, C++, Java, Fortran, Ruby, and BASIC, the "log" function returns the natural logarithm. The base-10 function, if it is available, is generally "log10."

• Some people use Log(x) (capital L) to mean log₁₀(x), and use log(x) with a lowercase l to mean log_e(x).

• The notation Log(x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.

• In some European countries, a frequently used notation is ^blog(x) instead of log_b(x).

This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/x, and having a simple definition), while the base 10 logarithms, or decimal logarithms, were more convenient for speeding calculations (back when they were used for that purpose). Thus natural logarithms were only extensively used in fields like calculus while decimal logarithms were widely used elsewhere.

In computer science, the base 2 logarithm is sometimes written as lg(x), as suggested by Edward Reingold and popularized by Donald Knuth. However, lg(x) is also sometimes used for the common log, and lb(x) for the binary log. In Russian literature, the notation lg(x) is also generally used for the base 10 logarithm. In German, lg(x) also denotes the base 10 logarithm, while sometimes ld(x) or lb(x) is used for the base 2 logarithm.

The clear advice of the United States Department of Commerce National Institute of Standards and Technology is to follow the ISO standard Mathematical signs and symbols for use in physical sciences and technology, ISO 31-11:1992, which suggests these notations:

• The notation "ln(x)" means log_e(x);
• The notation "lg(x)" means log₁₀(x);
• The notation "lb(x)" means log₂(x).